Principal Component Analysis

Principal component analysis (PCA) will be used as a dimensionality reduction technique to find the over-arching dimensions that represent knowledge about social relationships. In this study, we will replicate a previous study done nearly 40 years about (Wish et al.,1976).

Import data

This dataset was collected from a survey hosted on mturk. The survey data was cleaned with a separate python script. A matrix was created for the average rating of social relationships on dimensions that are thought to characterize these relationships. The relationships and dimensions used in this analysis are the same ones from Wish et al., 1976.

‘Wish relationships rated on Wish dimensions’

Quantitatively selecting the number of components

PCA will output the same number of components as there are dimension inputs. As the components are ranked by how much variance they explain, we can exclude some components which do not add much additional information.

Parallel Analysis

We will use parallel analysis to indicate what the optimal number of components to include would be.

## Parallel analysis suggests that the number of factors =  NA  and the number of components =  2
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## Parallel analysis suggests that the number of factors =  NA  and the number of components =  2

Parallel analysis indicates that having only 2 components would be optimal. But to better match the literature, and to be consistent with possible future analyses, we will include 4 components.

Screeplot

PCA with no rotation is done here to visualize the amount of variance accounted for by each component.

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##   2

PCA with varimax rotation

Rotations are used in principal component analyses to be able to better interpret the data. There are two main types of rotations, varimax and oblimin. Here, we will use varimax rotation, as it will maximize the component loadings so that dimensions are more strongly loaded onto a single component, rather than across components. Because of this, our resulting components may correlate with each other. Oblimin rotation results in components that are uncorrelated to each other.

## [1] "First four components account for 94.36% of the variance"

Component Loadings

## Component 1 highest positive loadings: Altruistic vs Selfish, Compatible vs incompatible goals and desires, Cooperative vs Competitive, Democratic vs Autocratic, Easy vs Difficult to resolve conflicts with each other, Emotionally close vs distant, Fair vs Unfair, Flexible vs Rigid, Friendly vs Hostile, Harmonious vs Clashing, Important vs Unimportant to individuals involved, Important vs Unimportant to society, Productive vs Destructive, Relaxed vs Tense, Sincere vs InsincereNULL
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## Component 1 highest negative loadings: NULL
## 
## Component 2 highest positive loadings: Difficult vs Easy to break off contact with each other, Emotional vs Intellectual, Emotionally close vs distant, Informal vs Formal, Intense vs Superficial feelings toward each other, Intense vs Superficial interaction with each other, Interesting vs Dull, Pleasure vs Work orientedNULL
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## Component 2 highest negative loadings: NULL
## 
## Component 3 highest positive loadings: Democratic vs Autocratic, Equal vs Unequal, Similar vs Different roles and behaviorNULL
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## Component 3 highest negative loadings: NULL
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## Component 4 highest positive loadings: Active vs Inactive, Important vs Unimportant to individuals involved, Important vs Unimportant to society, Intense vs Superficial interaction with each other, Interesting vs DullNULL
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## Component 4 highest negative loadings: NULL

Comparison to Wish Dimensions

Wish et al., 1976 - Table 2

Interactive Relationship Plots